Riemannian geometry, fall 2010

Lecturer

Scope

10 cu.

Type

Advanced studies

Prerequisites

Vector analysis, topology, linear algebra, some knowledge on smooth manifolds
(like the course "Johdatus diferentiaaligeometriaan") and on differential equations.

Lectures

Weeks 36-42 and 44-50, Tue 12-14, Thur 10-12 in room C123.

Exercise groups

Friday, 12-14 in room C124.

Exams

The course can be passed by an exam. Extra credit from home work classes.

Content

Differentiable manifolds (brief review)
Riemannian metrics
Connections
Geodesics
Curvature
Jacobi fields
Curvature and topology
Comparison geometry
The sphere theorem

Bibliography

DoCarmo: Riemannian geometry, Birkhäuser, 1992
Lee: Riemannian manifolds, An Introduction to Curvature, Springer, 1997

and (hand written) lecture notes: Holopainen: Differential Geometry (1999, 2001).

Lecture notes (first 101 pages, not proof-read): RiemannGeo.pdf

Registration

Did you forget to register? What to do.

Home work assignments and solutions

Exercise 1.
Solutions

Exercise 2.
Solutions

Exercise 3.
Solutions

Exercise 4.
Solutions

Exercise 5. (Corrected version)
Solutions

Exercise 6.
Solutions

Exercise 7. (Corrected version)
Solutions

Exercise 8.
Solutions

Exercise 9.
Solutions

Exercise 10.
Solutions

Exercise 11.
Solutions